Just perusing the Wikipedia articles on the religions of the RF and USA, it appears the non-religious numbers are somewhat comparable. If the Soviet Union was painted as so atheist and ungodly, why aren't there more atheists? Were those claims just inaccurate cold war propaganda?
I can't speak to "why aren't there more atheists," but I can try to answer "why are there so many Orthodox Christians." Here goes!
As I’m sure you’re aware, the Bolshevik Revolution ended up overthrowing the czarist regime; the Orthodox Chuch was then knocked from its place as the major Russian religion and replaced with Marxist materialism, later refined by Stalin. After the revolution, you have Stalin, Trotsky, and Bukharin moving to the head of the government, Stalin consolidating his power through lower politburos, purging the government, et cetera. I’m giving you this history so you can understand the form of atheism that was present in the Soviet Union, particularly during Stalin’s reign.
Stalin appended to Marxism and Leninism two tenets: (1) dialectical materialism, the philosophy that the world has objective reality independent of the human mind or spirit (in order to justify Stalin’s policies/attitude toward Russia at large), and (2) the cult of personality (COP), a quasi-religion which deified Lenin and which was dedicated to Stalin as leader/martyr for the cause of the Communist (i.e, Stalinist) uprising.
In 1917, when the Bolsheviks (under Lenin) came into power, they had a political agenda which, among other things, was concerned with Russian culture, with special emphasis on religion, which Peris terms “the metamorphosis of Holy Russia in to an atheistic Soviet Russia.” The Bolsheviks were dedicated to wiping out all systematized religion in Russia (making way for secular materialism), but the Orthodox Church was the main target because it occupied a central place in Russian culture. This was accomplished through the independent-of-the-party-on-paper League of Militant Atheism, formed as a propaganda machine to implement the Bolshevik plan to wipe out systematized religion.
During Destalinization (which took place directly following Stalin’s death), led by Kruschev and later by Gorbachev, the cult of personality which had become the dominant “religion” in Russia, was phased out gradually. This began with the renaming of cities like Stalino and Stalingrad (to Donetsk and Volgograd, respectively) and continued during Gorbachev’s era with the tearing down of monuments dedicated to Stalin and the disinterment of his body from Lenin’s mausoleum, into which the living Stalin had arranged to have his body placed and which solidified his COP hold on quasi-divinity and martyrdom. Destalinization efforts attacked the heart of the problem of the COP, but even today his name’s still pretty sacrosanct in the former Soviet Union.
The fact that all of this had to happen after the death of Stalin reinforces the magnitude of his religious influence; once Stalin was removed spiritually and physically, from the Soviet Union, the Union was on its way to collapse.
To answer your question directly, the Orthodox Church has always been integral to Russian national identity; this identity originates from the Rurik dynasty and its success at expanding and governing territory, and the alliances formed between the aristocracy and the monarchy. This alliance was “embedded in a range of myths, symbols, rituals, and institutions,” which indicate the ruling class’s effect on Russian national identity.
Following that chain, you have the Orthodox Church, inextricably intertwined with the czarship, to the point that the czar was seen as a religious, quasi-divine figure and held a place in the ikons of an average Russian home alongside Christ and the Virgin Mary. The Orthodox Church itself served as a facet of Russian national identity because of its ties to the ruling class and the military; it was uniquely suited to “fuse political and cultural national identity” because of its close ties with the czarship (Lieven).
So, quite predictably, post-Destalinization Russia has begun to return to the roots of its national/religious identity (Orthodox Church), which is closely tied to Putin in modern times. The quasi-religion which formed around Stalin fell with his death and has been replaced with the church that came before him, as Russia tried to regain its former identity.
Sources:
Froese, Paul. “Forced Secularization in Soviet Russia: Why an Atheistic Monopoly Failed”
Lieven, Dominic. “Russian, Imperial, and Soviet Identities”
Peris, Daniel. Storming the Heavens: The Soviet League of the Militant Godless
To piggyback on this, why do some other former Eastern Bloc countries have such high numbers of atheists, such as East Germany, the Czech Republic, and Estonia, as opposed to other ones, like Russia and Serbia?